The general rule is that if it ends in a vowel, the penultimate vowel gets the accent. If it ends in a noun, the ultimate vowel gets the accent. To that end, I am not aware of any difference between nouns and adjectives.
In Spanish, you wouldn't add the "an" to turn it into an adjective; rather, you'd say "del Caribe."
So I'm inclined to think that the accent should remain where the Spaniards would put it, despite the "an" the English tacked onto it, since the Spaniards would never add that "an."